Cellular Algebras and Frobenius Extensions Arising from Two-Parameter Permutation Matrices

Abstract

Let n be a positive integer, let R be a (unitary associative) ring, and let Mn(R) be the ring of all n by n matrices over R. For a permutation σ in the symmetry group Σn and a ring automorphism φ of R, we introduce the definition of σ - φ permutation matrices. The set Bn(σ, φ, R) of all σ - φ permutation matrices is proved to be a subring of Mn(R). It is shown that the extension Bn(σ, φ, R) ⊆ Mn(R) is a separable Frobenius extension. Moreover, if R is a commutative cellular algebra over the invariant subring Rφ of R, then Bn(σ, φ, R) is also a cellular algebra over Rφ.